Existence of weak solutions for a class of nonuniformly nonlinear elliptic equations in unbounded domains

The goal of this paper is to study the existence of non-trivial weak solutions for the nonuniformly nonlinear elliptic equation − div ( h ( x ) ∇ u ) + q ( x ) u = f ( x , u ) in an unbounded domain Ω ⊂ R N ( N ≥ 3 ) , where h ( x ) ∈ L loc 1 ( Ω ) . The solutions will be obtained in a subspace of the Sobolev space H 0 1 ( Ω ) and the proofs rely essentially on a variation of the Mountain pass theorem in [D.M. Duc, Nonlinear singular elliptic equations, J. London. Math. Soc. 40 (2) (1989) 420–440].